Find the general solution to the following second

Find the solution of the following Differential Equations
${\displaystyle (2x+e^y)dx+x{e}^{y}dy=0}$

I am looking to solve the following equations numerically:
$ax=\frac{d}{dt}(f(x,y,t)\frac{dy}{dt}),by=\frac{d}{dt}(g(x,y,t)\frac{dx}{dt})$
For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.
My best attempt so far is the following:
$\begin{array}{rl}{z}_1& =f(x,y,t)\frac{dy}{dt}\\ z_2& =g(x,y,t)\frac{dx}{dt}\\ z_3& =ax\\ z_4& =by\end{array}$
${\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right)}^{\prime }=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& \frac{a}{g(t,x,y)}& 0& 0\\ \frac{b}{f(t,x,y)}& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right)$
However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?

Use the definition of Laplace Transforms to show that:
${\displaystyle L\left\{t^n\right\}=\frac{n!}{s^{n+1}},n=1,2,3,\dots }$

Does ODE initial value problem produce beat or resonance phenomenon?
${\displaystyle x{}^{″}+9x=\sin \left(3t\right)}$
${\displaystyle x\left(0\right)=x^{\prime }\left(0\right)=0}$.
We are allowed to solve differential equations with TI-89.

Find the general solution to the differential equation
${\displaystyle \frac{dy}{dt}=t^3+2t^2-8t}$
Also, part 8B. asks: Show that the constant function ${\displaystyle y\left(t\right)=0}$ is a solution.
Ive

Use Laplace transform to solve the initial-value problem
${\displaystyle y=3,y\left(0\right)=1,y^{\prime }\left(0\right)=1}$

Use linear approximation, i.e. the tangent line, to approximate ${11.2}^2$ as follows :
Let $f(x)=x^2$ and find the equation of the tangent line to $f(x)$ at $x=11$. Using this, find your approximation for ${11.2}^2$.